Recently we published one application in Google Play. This is our first experience. But my manager asked me to check whether our application name has already been taken by someone/whether the name has a patent. This seems wired to me.

So, I did a search and come up with one information, like Until your application get famous and earning lots of money, ignore about patent. No body will attack you.

But I don't know exactly which apps names are under patent. How can I find the list of all patent apps? Or Google will send us some information if the name has already been patent etc.

Trademarks aren't like patents, there isn't a single national registration. Someone could obtain a federal trademark (search at tess2.uspto.gov), but there are also state registrations, as well as unregistered trademarks.

If you are going to be spending real money on the mark, it's worthwhile to engage an attorney to do a full analysis.